DOCSLIB.ORG

Tuning with Just Intonation and Equal Temparement

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tuning with Just Intonation and Equal Temparement Tuning With Just Intonation and Equal Temparement Jacob Scott December 9, 2015 1 A Brief Introduction The discovery that the freqency of a vibrating string is inversely proportional to its length is generally credited to the Greek polymath Pythagoras in the late 6th century BC [1]. The implication of this discovery was that for any string of length l vibrating at frequency λ hertz, a 1 string of length 2 l will vibrate at 2λ hertz. Doubling the frequency in this way (which can be done by artificially creating a node exactly half way along the string) gives the first harmonic in the harmonic 1 series. The second harmonic would be a string of length 3 l vibrating at 3λ hertz, and so on [4] (Note: from this point the stan- dard ratio/fraction systems will be fundamental frequency used to denote pitches in just into- nation. For more information, see [5].) Pythagoras found that by repeatedly 1st harmonic (octave) taking the interval between the 1st and 2nd harmonics (i.e. a perfect fifth), all 12 notes of the chromatic scale could be found (for more ex- 2nd harmonic (Perfect/Pythagorian fifth) planation see the cycle of fifths [2]). However, completing this cycle using 'true' harmonics, the final frequency 3rd harmonic (second octave) of the cycle was found to be ’off' from the original, and to compen- sate, a highly dissonant 'wolf fifth’ had to be used to complete the oc- 4th harmonic (major third in second octave) tave. (See https://www.youtube. com/watch?v=Dl7iIzvUMGg). Many different methods of tuning have been invented to try and solve (for sound, see https://soundcloud.com/jakescottmusic/harmonics) the problem of this apparent error in the fundamental mechanics of music. Figure 1: The first four harmonics Broadly, they can be categorised un- der equal temperament and systems of just intonation. 2 Equal Temperament The modern system of tuning simply divides the octave into 12 equal steps. A formula for this can be derived as follows: If Pn is a known frequency and Pa is exactly one octave higher than Pn, then Pn = 2Pa. 12 Since differences in frequencyp are not additive but multiplicative, Pn = Pax Therefore x12 = 2 =) x = 12 2 1 p 12 12 And from this, Pn = Pa( 2) p 12 n If the frequency of Pa is known, then Pn can be found using the formula Pn = Pa( 2) . Moreover, if Pn and Pa are n and a semitones from some other 'base' frequency, respectively, then we can use the following formula: p 12 n−a Pn = Pa( 2) 2.1 Cents 1 The cent is a logarithmic unit in music theory, denoting 100 th of a semitone. The advantage of using cents is that they can be used additively to find the difference between two frequencies, which gives a much clearer and more intuitive representation of small intervals. If we take the previous formula, with a = 0, we can derive the formula for the cent. p 12 n Pn n Pn n Pn Pn = Pa( 2) =) = 2 12 =) log2( ) = log22 =) n = 12log2( ) P0 P0 12 P0 Multiplying by 100 to give cents we obtain: Pn c = 1200log2( ) P0 3 Just Intonation Just intonation is any system of tuning where intervals are given by ratios of 'small' positive integers. Many systems of just intonation have been devised that fit this definition, but for the purposes of this project I will be using five-limit just intonation, and particularly assymetric five-limit just intonation. For more information on types of just intonation, see [5] and [3]. 3.1 Five-limit Just Intonation (Note: the following can be found in greater detail online, see [3]) The fractions in Table 1 are found as fol- lows: Factor 1/9 1/3 1 3 9 For any given note, the corresponding fac- 5 Note D− A E B F]+ tors are multiplied together to give the the Ratio 10/9 5/3 5/4 15/8 45/32 correct fraction for that note at some oc- 1 Note B[ F C G D tave. To obtain the note in its correct oc- Ratio 16/9 4/3 1 3/2 9/8 tave, the fraction is multiplied by 2 to a 1/5 Note G[− D[− A[ E[ B[ chosen integer power. Ratio 64/45 16/15 8/5 6/5 9/5 1 1 6 In the case of E[, 5 · 3 · 2 = 5 . In Table 1, notes with enharmonic equiva- Table 1: Fifteen Pitches lents (i.e, two notes that whilst occupying the same position on the keyboard, do not have the same exact frequency) have been shaded with matching colours. To obtain five-limit tuning, the shaded frequencies on the left of the table (i.e. the notes corresponding to fractions 10/9, 16/9 and 64/45) are discarded, giving a full 12-tone justly intonated scale. (Note: there are other systems for eliminating enharmonic equivalents, this one in particular gives asymmetric five-limit tuning.) Sagemath can be used to create a graph showing the differences in cents between equal-tempered and justly intonated pitches, as shown in Figure 2. The complete code used to create the graph can be found at https://cloud.sagemath.com/projects/bd250a4b-64e4-4530-aa25-0d15b6d41b61/ files/pitch%20graph.sagews 2 (for an aural comparison see https://soundcloud.com/jakescottmusic/equal-temperament-just-intonation) Figure 2: Difference in cents between justly intonated and equal-tempered pitches 4 Tuning in Python The following code shows how python can be used to calculate equal-tempered and justly intonated fre- quencies, and the difference between them in cents. The program the code is taken from has the primary purpose of using randomly generated chords to highlight the differences in these tuning systems, though it also offers the ability to tune chords inputted by a user. The chord Cm7 (made up of notes C, E[, G and B[) in the key of C particularly demonstrates the difference between just intonation and equal tempera- ment. (See Figure 2: E[ and B[ correspond to semitones 3 and 10 respectively - points at which the blue line moves particularly far from the red.) The full code can be found at https://cloud.sagemath.com/ projects/bd250a4b-64e4-4530-aa25-0d15b6d41b61/files/tuner%20final%20program.sagews, but note that the python module 'pyo' must be installed in order to generate audio using the program. freqs_equal = [] # creates an empty list for the final equal-tempered frequencies to be added to for note in notelist: letter = note[0] # separates 'note' element into its octave and the note's name. octave = note[1] freq = 440 * ((2 ** (1.0/12.0)) ** (letter - 9 - (12 * 4))) # calculates note in its lowest octave, relative to the frequence of A4 (440 Hz) if octave != 0: freq = freq * ((2 ** (1.0/12.0)) ** (octave * 12)) # transposes the note to its correct octave freq = int(round(freq, 0)) # rounds up freq and turns it into an int freqs_equal.append(freq) # adds the found frequency to the list self.freqs_equal = freqs_equal freqs_just = [] for note in notelist: letter = note[0] octave = note[1] # get the note's name and octave if letter - key >= 0: interval = letter - key # calculates interval else: interval = (letter + 12) - key octave -= 1 # calculates interval for a different octave, changes octave to compensate cents = [] #creates an empty list for e in range(0, len(notelist)): #do this as many times as there are notes in the chord justfreq = freqs_just[e] equalfreq = freqs_equal[e] difference = 1200 * math.log((float(justfreq) / float(equalfreq)), 2) # uses formula to find difference in cents between two frequencies if difference < 0: 3 difference = 1200 * math.log((float(equalfreq) / float(justfreq)), 2) # if the difference is negative, do it the other way round difference = int(round(difference, 0)) # rounds up difference and turns it into an int cents.append(difference) # add the difference to the list self.cents = cents 5 Conclusions In this report I have briefly explained the difference between equal-temperament and just intonation, giving examples of how the differences can be represented both visually and aurally using sagemath and python. This project could be developed further by using a specifically music-orientated programming language such as ChucK to investigate how the tuning capabilities of computers can be used practically in areas such as composition and performance. In addition to this, many non-Western musical cultures do not split the octave into 12 notes as we do. Exploring how the harmonic series relates to tuning systems such as these could potentially offer much more insight into the mathematics of tuning. References [1] John H Chalmers. Divisions of the tetrachord: a prolegomenon to the construction of musical scales. Frog Peak Music, 1993. [2] Wikipedia. Circle of fifths | wikipedia, the free encyclopedia, 2015. [Online; accessed 9-December- 2015]. [3] Wikipedia. Five-limit tuning | wikipedia, the free encyclopedia, 2015. [Online; accessed 6-December- 2015]. [4] Wikipedia. Harmonic series (music) | wikipedia, the free encyclopedia, 2015. [Online; accessed 5-December-2015]. [5] Wikipedia. Just intonation | wikipedia, the free encyclopedia, 2015. [Online; accessed 5-December- 2015]. 4.
Load more

Recommended publications
  • 3 Manual Microtonal Organ Ruben Sverre Gjertsen 2013
    3 Manual Microtonal Organ http://www.bek.no/~ruben/Research/Downloads/software.html Ruben Sverre Gjertsen 2013 An interface to existing software A motivation for creating this instrument has been an interest for gaining experience with a large range of intonation systems. This software instrument is built with Max 61, as an interface to the Fluidsynth object2. Fluidsynth offers possibilities for retuning soundfont banks (Sf2 format) to 12-tone or full-register tunings. Max 6 introduced the dictionary format, which has been useful for creating a tuning database in text format, as well as storing presets. This tuning database can naturally be expanded by users, if tunings are written in the syntax read by this instrument. The freely available Jeux organ soundfont3 has been used as a default soundfont, while any instrument in the sf2 format can be loaded. The organ interface The organ window 3 MIDI Keyboards This instrument contains 3 separate fluidsynth modules, named Manual 1-3. 3 keysliders can be played staccato by the mouse for testing, while the most musically sufficient option is performing from connected MIDI keyboards. Available inputs will be automatically recognized and can be selected from the menus. To keep some of the manuals silent, select the bottom alternative "to 2ManualMicroORGANircamSpat 1", which will not receive MIDI signal, unless another program (for instance Sibelius) is sending them. A separate menu can be used to select a foot trigger. The red toggle must be pressed for this to be active. This has been tested with Behringer FCB1010 triggers. Other devices could possibly require adjustments to the patch.
    [Show full text]
  • Download the Just Intonation Primer
    THE JUST INTONATION PPRIRIMMEERR An introduction to the theory and practice of Just Intonation by David B. Doty Uncommon Practice — a CD of original music in Just Intonation by David B. Doty This CD contains seven compositions in Just Intonation in diverse styles — ranging from short “fractured pop tunes” to extended orchestral movements — realized by means of MIDI technology. My principal objectives in creating this music were twofold: to explore some of the novel possibilities offered by Just Intonation and to make emotionally and intellectually satisfying music. I believe I have achieved both of these goals to a significant degree. ——David B. Doty The selections on this CD pro­­cess—about synthesis, decisions. This is definitely detected in certain struc- were composed between sampling, and MIDI, about not experimental music, in tures and styles of elabora- approximately 1984 and Just Intonation, and about the Cageian sense—I am tion. More prominent are 1995 and recorded in 1998. what compositional styles more interested in result styles of polyphony from the All of them use some form and techniques are suited (aesthetic response) than Western European Middle of Just Intonation. This to various just tunings. process. Ages and Renaissance, method of tuning is com- Taken collectively, there It is tonal music (with a garage rock from the 1960s, mendable for its inherent is no conventional name lowercase t), music in which Balkan instrumental dance beauty, its variety, and its for the music that resulted hierarchic relations of tones music, the ancient Japanese long history (it is as old from this process, other are important and in which court music gagaku, Greek as civilization).
    [Show full text]
  • Shifting Exercises with Double Stops to Test Intonation
    VERY ROUGH AND PRELIMINARY DRAFT!!! Shifting Exercises with Double Stops to Test Intonation These exercises were inspired by lessons I had from 1968 to 1970 with David Smiley of the San Francisco Symphony. I don’t have the book he used, but I believe it was one those written by Dounis on the scientific or artist's technique of violin playing. The exercises were difficult and frustrating, and involved shifting and double stops. Smiley also emphasized routine testing notes against other strings, and I also found some of his tasks frustrating because I couldn’t hear intervals that apparently seemed so familiar to a professional musician. When I found myself giving violin lessons in 2011, I had a mathematical understanding of why it was so difficult to hear certain musical intervals, and decided not to focus on them in my teaching. By then I had also developed some exercises to develop my own intonation. These exercises focus entirely on what is called the just scale. Pianos use the equal tempered scale, which is the predominate choice of intonation in orchestras and symphonies (I NEED VERIFICATION THAT THIS IS TRUE). It takes many years and many types of exercises and activities to become a good violinist. But I contend that everyone should start by mastering the following double stops in “just” intonation: 1. Practice the intervals shown above for all possible pairs of strings on your violin or viola. Learn the first two first, then add one interval at a time. They get harder to hear as you go down the list for reasons having to do with the fractions: 1/2, 2/3, 3/4, 3/5, 4/5, 5/6.
    [Show full text]
  • Playing Music in Just Intonation: a Dynamically Adaptive Tuning Scheme
    Karolin Stange,∗ Christoph Wick,† Playing Music in Just and Haye Hinrichsen∗∗ ∗Hochschule fur¨ Musik, Intonation: A Dynamically Hofstallstraße 6-8, 97070 Wurzburg,¨ Germany Adaptive Tuning Scheme †Fakultat¨ fur¨ Mathematik und Informatik ∗∗Fakultat¨ fur¨ Physik und Astronomie †∗∗Universitat¨ Wurzburg¨ Am Hubland, Campus Sud,¨ 97074 Wurzburg,¨ Germany Web: www.just-intonation.org [email protected] [email protected] [email protected] Abstract: We investigate a dynamically adaptive tuning scheme for microtonal tuning of musical instruments, allowing the performer to play music in just intonation in any key. Unlike other methods, which are based on a procedural analysis of the chordal structure, our tuning scheme continually solves a system of linear equations, rather than relying on sequences of conditional if-then clauses. In complex situations, where not all intervals of a chord can be tuned according to the frequency ratios of just intonation, the method automatically yields a tempered compromise. We outline the implementation of the algorithm in an open-source software project that we have provided to demonstrate the feasibility of the tuning method. The first attempts to mathematically characterize is particularly pronounced if m and n are small. musical intervals date back to Pythagoras, who Examples include the perfect octave (m:n = 2:1), noted that the consonance of two tones played on the perfect fifth (3:2), and the perfect fourth (4:3). a monochord can be related to simple fractions Larger values of mand n tend to correspond to more of the corresponding string lengths (for a general dissonant intervals. If a normally consonant interval introduction see, e.g., Geller 1997; White and White is sufficiently detuned from just intonation (i.e., the 2014).
    [Show full text]
  • A Study of Microtones in Pop Music
    University of Huddersfield Repository Chadwin, Daniel James Applying microtonality to pop songwriting: A study of microtones in pop music Original Citation Chadwin, Daniel James (2019) Applying microtonality to pop songwriting: A study of microtones in pop music. Masters thesis, University of Huddersfield. This version is available at http://eprints.hud.ac.uk/id/eprint/34977/ The University Repository is a digital collection of the research output of the University, available on Open Access. Copyright and Moral Rights for the items on this site are retained by the individual author and/or other copyright owners. Users may access full items free of charge; copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational or not-for-profit purposes without prior permission or charge, provided: • The authors, title and full bibliographic details is credited in any copy; • A hyperlink and/or URL is included for the original metadata page; and • The content is not changed in any way. For more information, including our policy and submission procedure, please contact the Repository Team at: [email protected]. http://eprints.hud.ac.uk/ Applying microtonality to pop songwriting A study of microtones in pop music Daniel James Chadwin Student number: 1568815 A thesis submitted to the University of Huddersfield in partial fulfilment of the requirements for the degree of Master of Arts University of Huddersfield May 2019 1 Abstract While temperament and expanded tunings have not been widely adopted by pop and rock musicians historically speaking, there has recently been an increased interest in microtones from modern artists and in online discussion.
    [Show full text]
  • Exploration of an Adaptable Just Intonation System Jeff Snyder
    Exploration of an Adaptable Just Intonation System Jeff Snyder Submitted in partial fulfillment of the requirements of the degree of Doctor of Musical Arts in the Graduate School of Arts and Sciences Columbia University 2010 ABSTRACT Exploration of an Adaptable Just Intonation System Jeff Snyder In this paper, I describe my recent work, which is primarily focused around a dynamic tuning system, and the construction of new electro-acoustic instruments using this system. I provide an overview of my aesthetic and theoretical influences, in order to give some idea of the path that led me to my current project. I then explain the tuning system itself, which is a type of dynamically tuned just intonation, realized through electronics. The third section of this paper gives details on the design and construction of the instruments I have invented for my own compositional purposes. The final section of this paper gives an analysis of the first large-scale piece to be written for my instruments using my tuning system, Concerning the Nature of Things. Exploration of an Adaptable Just Intonation System Jeff Snyder I. Why Adaptable Just Intonation on Invented Instruments? 1.1 - Influences 1.1.1 - Inspiration from Medieval and Renaissance music 1.1.2 - Inspiration from Harry Partch 1.1.3 - Inspiration from American country music 1.2 - Tuning systems 1.2.1 - Why just? 1.2.1.1 – Non-prescriptive pitch systems 1.2.1.2 – Just pitch systems 1.2.1.3 – Tempered pitch systems 1.2.1.4 – Rationale for choice of Just tunings 1.2.1.4.1 – Consideration of non-prescriptive
    [Show full text]
  • A Whole Different Serving of Tapioca: Exploring Just Intonation Dissonance
    SPRING 2002 1 1 Volume 11, Number 1 THE JOURNAL OF THE JUST INTONATION NETWORK A PROJECT OF OTHER MUSIC, INC. A Whole Different Serving of Tapioca: Exploring Just Intonation Dissonance by Kyle Gann wrote a song, “Jargon,” completely in seconds, sevenths, Just Intonation dissonance, said Harry Partch in one of and ninths, and justified them with a satirical text (“Let tuning theory’s more ringing phrases, is “a whole dif- horrid Jargon split the air/And rive the Nerves asunder”). ferent serving of tapioca” from equally tempered dis- Somewhat analogously, I found a text and allowed pro- sonance. I had no idea what that meant when I first heard grammatic considerations to lead me into several differ- it. I’ve now used enough Just Intonation dissonance in ent kinds and degrees of just dissonance. my music to learn how right he was. My text was taken from My Life on the Plains, the My early Just Intonation works were rather fanatically autobiographical memoir of General George Armstrong consonant. Having expended years of time and energy Custer. 1 If ever a text cried out for dissonance, it was to achieve pure consonance, I was at a loss as to how to this, bristling with unacknowledged contradictions. One justify returning to dissonance, and as to what strategy or moment, Custer would admit that if he were an Indian, he theory to use as a guideline. After all, if you start working would ignore U.S. Government mandates and go live on in Just Intonation to get rid of beats in your sonic surface, the plains; a few pages later he would damn all Indians under what theoretical aegis do you bring beats back in? as brutes only governable by force.
    [Show full text]
  • Tuning, Tonality, and 22-Tone Temperament
    Tuning, Tonality, and Twenty-Two-Tone Temperament Paul Erlich ([email protected]) (originally published in Xenharmonikôn 17, Spring 1998; revised 4/2/02) Introduction Western modal and tonal music generally conforms to what is known as the 5-limit; this is often explained (e. g., by Hindemith) by stating that the entire tuning system is an expression of integer frequency ratios using prime factors no higher than 5. A more correct description is that all 5-limit intervals, or ratios involving numbers no higher than 5, aside from factors of 2 arising from inversion and extension, are treated as consonances in this music, and thus need to be tuned with some degree of accuracy. The diatonic scale of seven notes per octave has been in use since ancient Greek times, when harmonic simultaneities (other than unisons and octaves) were not used and the only important property of the scale was its melodic structure. As melodic considerations continued to play a dominant role in Western musical history, scales other than the diatonic had very limited application, and harmonic usage was intimately entangled with diatonic scale structure. It is fortuitous that the diatonic scale allowed for many harmonies that displayed the psychoacoustic phenomenon of consonance, defined by small- integer frequency ratios, as well as many that did not. Medieval music conformed to the 3-limit, the only recognized consonances being the octave (2:1) and the perfect fifth (3:2) (plus, of course, their octave inversions and extensions, obtained by dividing or multiplying the ratios by powers of 2; this “octave equivalence” will be implicit from here on.1) The diatonic scales were therefore tuned as a chain of consecutive 3:2s (Pythagorean tuning).
    [Show full text]
  • It's Just Intonation
    It’s Just Intonation Beth Bronk It is faulty intonation that creeps in and spoils what would otherwise be a fine performance . good intonation is the indispensable of the mechanics of music. —William Revelli, 1938 The foundation of all good intonation is to have strong inner hearing, a clear plan or desire for the exact coming pitch. (Joukamo-Ampuja and Wekre, 1999, p. 47). We Already Teach that most closely match Equal training for better intonation. Fundamental Skills – Temperament tuning. This adjusts Performers and teachers must Intonation is Fundamental, Too! the instruments to factory settings have some very practical knowledge As band directors, we and allows all the members of the and skills in order to successfully recognize the importance of ensemble to start from A=440. teach intonation. These include: taking a fundamental approach 2. Students and teachers must • The ability to produce (or to performance. We would all strive constantly for excellence in teach our students to produce) probably agree that the fastest way tone quality. a vibrant, centered, characteristic to make an ensemble sound better 3. Performers must develop the tone quality is to teach each student to perform ability to audiate. • Knowledge about the with good characteristic tone “Audiation”, or inner hearing, intonation tendencies of specific quality. Next to good tone quality, is a term originally coined by Ed notes on instruments improving intonation is probably Gordon. “…When you are listening • The ability to correctly adjust the next fastest way to make our to music, you are giving meaning the length of the instrument ensembles sound better.
    [Show full text]
  • MTO 12.3: Duffin, Just Intonation in Renaissance Theory and Practice
    Volume 12, Number 3, October 2006 Copyright © 2006 Society for Music Theory Ross W. Duffin ABSTRACT: Just intonation has a reputation as a chimerical, theoretical system that simply cannot work in practice. This is based on the assessment of most modern authorities and supported by misgivings expressed during the Renaissance when the practice was supposedly at its height. Looming large among such misgivings are tuning puzzles printed by the 16th-century mathematician, Giovanni Battista Benedetti. However, Renaissance music theorists are so unanimous in advocating the simple acoustical ratios of Just intonation that it seems clear that some reconciliation must have occurred between the theory and practice of it. This article explores the basic theory of Just intonation as well as problematic passages used to deny its practicability, and proposes solutions that attempt to satisfy both the theory and the ear. Ultimately, a resource is offered to help modern performers approach this valuable art. Received June 2006 Introduction | Theoretical Background | Benedetti's Puzzles | Problematic Passages | Is Just Tuning Possible? A New Approach | Problem Spots in the Exercises | Rehearsal Usage | Conclusion Introduction The idea . that one can understand the ratios of musical consonances without experiencing them with the senses is wrong. Nor can one know the theory of music without being versed in its practice. [1] So begins the first of two letters sent by the mathematician Giovanni Battista Benedetti to the composer Cipriano de Rore in 1563. Subsequently publishing the letters in 1585,(1) Benedetti was attempting to demonstrate that adhering to principles of Just intonation, as championed most famously by Gioseffo Zarlino,(2) would, in certain cases, cause the pitch of the ensemble to migrate.
    [Show full text]
  • The Search for Pure Intonation
    Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings Intonation in Performance Practice: An Historical, Mathematical and Aural Comparison of Four Tunings Nick Nyderek, Music Dr. Matthew Faerber,Ph.D. Department of Music ABSTRACT This paper explores several different scale intonations and temperaments, including Pythagorean intonation, limit-5 just intonation, mean-tone temperament, and equal temperament. Historical context is provided as well as possible reasons for the development of new temperaments and intonations. These temperaments and intonations are compared mathematically and musically with pure intonation. A primary focus of the provision, for aural analysis and comparison, is a computer generated audio recording of each of the specified intonations and temperaments. There are also graphical representations comparing different temperaments and intonations. Introduction Since the time of Pythagoras, scale intonations have been an ongoing subject of study and refinement for musicians. Evolving musical styles, development of new instruments, and individual taste have all been components in the development of different intonations. For modern musicians and music educators, the importance of understanding different intonations and temperaments lies largely within performance practice. Tuning is a primary element of performance practice. For example, a vocal ensemble may not present a historically informed performance if the equal tempered tuning of a piano is the basis for the chosen intonation: equal temperament was not implemented in practice until a century or more after the Renaissance. Four prominent intonations and temperaments include Pythagorean intonation, just intonation, mean-tone temperament, and equal temperament. Each of these intonations represents a significant development in the history of intonation in western music.
    [Show full text]
  • The Classical Indian Just Intonation Tuning System
    The Classical Indian Just Intonation Tuning System with 22 SRUTI-s defining the 7 SWARA-s of Hindu Classical Music combining the three different kinds of SRUTI which are understood as PRAMANA ("measuring" or "standard") SRUTI = Syntonic Comma (81/80) = 21.5 cents NYUNA ("deficient") SRUTI = Minor Chroma (25/24) = 70.7 cents Wolfgang von Schweinitz October 20 - 22, 2006 PURNA ("fullfilling") SRUTI = Pythagorean Limma (256/243) = 90.2 cents (where PURNA SRUTI may also, enharmonically, be interpreted as the sum of PRAMANA SRUTI and NYUNA SRUTI = Major Chroma (135/128 = 81/80 * 25/24) = 92.2 cents) purna nyuna purna nyuna purna nyuna purna purna pramana pramana pramana pramana pramana pramana pramana pramana pramana pramana nyuna purna nyuna purna nyuna nyuna # # # # # # " ! # " # # ! # # # <e e f m n >m n # # # # # f # n t e u f m n # # ! # # # <e e# f# m# n# e f m n >m ! n" t <f e u f m n # # # # # # " ! # " # # ! # # # <e e f m n >m n ! # # # # # f # n t e u f m n # # ! # # # <e e# f# m# n# n o u v >u n " t <f e u f m n # # " ! ! # # ! # # # <n n# o# u# v# >u n # # # # f # # n" t e u f m n # # # ! # # # <e e # f # m# n# n o u v >u n " t <f e u f m n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 # # ! # # ! <e e# f# m# n# >m n " # # f n" t e# u f# m# n# $ # ! <e e# f# m# n# n# o# u# v# >u n " t <f e#u f# m# n# Sa ri ri Ri Ri ga ga Ga Ga ma ma Ma Ma Pa dha dha Dha Dha ni ni Ni Ni Sa 1 25 21 256 135 16 10 9 7 32 6 5 81 4 27 45 64 729 10 3 25 128 405 8 5 27 7 16 9 15 243 40 2 Ratio 1 24 20 243 128 15 9 8 6 27 5 4 64 3 20 32 45
    [Show full text]